Name: Arithmetic Mean & Geometric Mean: Illustration
Uploaded: 2019-07-19
Duration: 2 min 43 s
Description: The arithmetic mean and geometric mean are related to each other. The arithmetic mean of two positive numbers can never be less than their geometric mean. Wa...

Question

If

A_{1}, A_{2}

be two AM's and

G_{1}, G_{2}

be two GM's between

a

and

b

, then find the value of

\frac{A_{1} + A_{2}}{G_{1} G_{2}}

Accepted Answer

It is given that $A_{1}$ and $A_{2}$ are the A.M.s between $a$ and $b$

$'n' arthmetic means between 'a' and ' b^{'} are as follows A_{1} = a + d = a + \frac{b - a}{n + 1} A_{2} = a + 2 d = a + \frac{2 (b - a)}{n + 1}$

$A_{3} = a + 3 d = a + \frac{3 (b - a)}{n + 1} A_{n} = a + n d = a + \frac{n (b - a)}{n + 1}$

Common difference $d = \frac{b - a}{n + 1}$

Thus, $a, A_{1}, A_{2}$ and $b$ are in A.P. with common difference $d$ .

Here, $d = \frac{b - a}{3}$

$∴ A_{1} = a + \frac{b - a}{3} = \frac{2 a + b}{3}$

and $A_{2} = a + \frac{2 (b - a)}{3} = \frac{a + 2 b}{3}$

It is also given that $G_{1}$ and $G_{2}$ are the G.M.'s between $a$ and $b$

Thus, $a, G_{1}, G_{2}$ and $b$ are in G.P. with common ratio $r$ .

Here, $r = {(\frac{b}{a})}^{\frac{1}{3}}$

$∴ G_{1} = a {(\frac{b}{a})}^{\frac{1}{3}} = b^{\frac{1}{3}} a^{\frac{2}{3}}$

and $G_{2} = a {[{(\frac{b}{a})}^{\frac{1}{3}}]}^{2} = b^{\frac{2}{3}} a^{\frac{1}{3}}$

$\Rightarrow \frac{A_{1} + A_{z}}{G_{1} G_{z}} = \frac{\frac{2 a + b}{3} + \frac{a + 2 b}{3}}{b^{\frac{1}{3}} a^{\frac{2}{3}} \times b^{\frac{2}{3}} a^{\frac{1}{3}}} = \frac{a + b}{a b}$

If $A_{1}, A_{2}$ be two AM's and $G_{1}, G_{2}$ be two GM's between $a$ and $b$ , then find the value of $\frac{A_{1} + A_{2}}{G_{1} G_{2}}$

Important Questions on Geometric Progressions

Learn and Prepare for any exam you want !

If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of A1+A2G1G2

Important Questions on Geometric Progressions

Learn and Prepare for any exam you want !

If $A_{1}, A_{2}$ be two AM's and $G_{1}, G_{2}$ be two GM's between $a$ and $b$ , then find the value of $\frac{A_{1} + A_{2}}{G_{1} G_{2}}$